Polytropic expansion

Efficiency for a turboexpander is calculated on the basis of isentropic rather than polytropic expansion even though its efficiency is not 100 percent. This is done because the losses are largely introduced at the discharge of the machine in the form of seal leakages and disk friction which heats the gas leaking past the seals and in exducer losses. (The exducer acts to convert the axial-velocity energy from the rotor to pressure energy.)... 

But another approach to multi-step cooling [8, 9] involves dealing with the turbine expansion in a manner similar to that of analysing a polytropic expansion. Fig. 4.4 shows gas flow (1 + ijj) at (p,T) entering an elementary process made up of a mixing process at constant pressure p, in which the specific temperature drops from temperature T to temperature T, followed by an isentropic expansion in which the pressure changes to (p dp) and the temperature changes from T to (7 - - dT). 

The two step cooling example given above can in theory be extended to multi-step cooling of the turbine. It is more convenient to treat the turbine expansion as a modification of normal polytropic expansion the analysis is essentially an adaptation of that given in Section 4.2.1.3 for the multi-step cooled turbine cycle. 

Process refrigeration is used at many different temperature levels to condense or cool gases, vapors, or liquids. Refrigeration is necessary when the process requires cooling to a temperature not reliably available from the usual water service or other coolant source, includingjoule-Thompson, or polytropic expansion of natural gas or process system vapors. 

The energy recoverable from the expansion of a gas can be estimated by assuming polytropic expansion see Section 3.13.2 and Example 3.17. 

The equations for work done and for heat added in the case of the general polytropic expansions are ... 

When the exponent on the ratio of the pressures is other than the ratio of the heat capacities, this is a general expression for a polytropic expansion. 

From equation 3.31, work done by gases as a result of polytropic expansion... 

The limiting case for a polytropic expansion where n = I needs to be treated sep>arately as the mathematics of the pmlytropic expansion lead to an indeterminate solution when n is precisely unity. 

Steady-state flow through a nozzle for a gas undergoing a polytropic expansion... 

As in the case of the general polytropic expansion, and for similar reasons, we may neglect the potential energy term. As before, we may also simplify this equation by setting the inlet velocity to zero for gas or vapour coming from a vessel or header, so that the velocity is given by ... 

Combining the two expressions (5.51) and (5.52) with the equation for the sonic velocity, (5.47), gives the downstream speed of sound in terms of the upstream conditions for a polytropic expansion ... 

Achieving sonic flow during an isothermal expansion is a highly improbable eventuality because the rapid heat transfer needed would take a great effort to contrive. Nevertheless, it is a theoretical possibility and has an interest as the limiting case of a reversible polytropic expansion. Accordingly a brief treatment will be provided here for completeness. 

The expander efficiency was calculated with aid of a polytropic expansion coefficient of 1.67, that is, an isotropic efficiency of 70%. A reversible process [22] is totally misleading for determining the potential of such a hybrid system. In addition, the mechanical and electrical transmission losses were evaluated with a transmission efficiency of 95%. 

Gas requirements were determined by measuring the pressure drop in the gas manifold bottles after they had returned to ambient temperature.. Cooling of the bottles is associated with the polytropic expansion of the gas within the bottles during pressurization of the storage tank. 

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